An Inside-Out Course on Number Theory (Pt 3)

Link to previous posts: Pt1 Pt2

I’ve been doing a lot of soul-searching about this Inside Out course lately because of a growing dissatisfaction that I’m having about the class and also that some students are expressing.

One of the things I really wanted to do was to create a mathematics class that rehumanizes rather than dehumanizes, and I think that while students are generally feeling very positive about the class, I do feel like I am slipping into prior habits about teaching mathematics.

In previous class sessions, we have been sitting in four groups of 4-5 students and I have been visibly randomly assigning students to the groups for the purpose of having students work with each other and recognize each others’ mathematical talents.

This past Friday, one of my students suggested that we sit in one large circle instead of smaller groups. The result of that was that students generally still worked with the people that they happened to sit next to (there were still about 5 6 clumps of people) but there was also a small number of students who were quicker than the rest in making mathematical connections who were talking across the whole room. That increased interaction added another layer to the small group interactions because it helped to spread ideas faster across the room. It also, however, had the effect of reinforcing a particular kind of mathematical competence in the classroom–the kind where mathematical competence is associated with being quick or fast at getting answers. And, that is something that I’m really trying hard to break away from–I want students to be able to see themselves as being mathematical brilliant in lots of different ways, not just being quick and fast.

I cannot blame this narrow view of mathematical competence on this large circle classroom arrangement. Certainly a lot of it comes from the larger society and the way that many of us learn mathematics in schools–we are taught to value speed and accuracy and to associate that as being smart. But, I also take responsibility in the way that I have been designing the tasks that students are doing.

The materials I’ve been creating for this Introduction to Number Theory are derived from the 2009 Park City Mathematics Institute Teacher Leadership Program morning mathematics class (book version). Because of the mathematical preparation level of the students in the course, I have removed many pieces of the original PCMI course materials so that the main thrust of the work that we’ve been doing has been to tabulate various number-theoretic functions, determining if they are multiplicative or not, and looking for patterns.

The problems in and of themselves, are not group worthy. They can be done by individuals working on their own, and they are presented in numerical order so there is a sense of completing problems in a sequence, one by one. The design of the materials, therefore, reinforces the very thing that I am trying to avoid: a perspective on mathematical competence that values speed and accuracy. Some students in the class get farther along than others in the course, and it’s become clear who those students are. Other students turn to those students for help, which is great, but I don’t see the reverse happening as often.

At the end of each class, I have been spending about 15 minutes in a whole-group discussion in which students share out their mathematical observations and questions, and give gratitude to each other. I was trying to steer students in pointing out the different ways that we all are mathematically competent. Most of the comments lately have been along the lines of “I’m really grateful to XX for explaining YY to me so clearly.”

All of this is really not germane to teaching a course in a prison; this kind of thing happens in math classrooms all around the world. I’m hyper-focused on creating a more humanizing course mainly because I’m teaching in what might be the ultimate dehumanizing environment: a prison. But, the reality is that we need to work toward more rehumanizing classrooms all around the world.

So, what to do? One way to make a classroom more rehumanizing is to listen to your students and find out what they want and need. I did that last Friday by asking them to answer three questions anonymously: (1) What mathematical connections are you still wondering about in this class? (2) What can I (your instructor) do to make this class more awesome? (3) What can you do to make this class more awesome?

Generally, students’ comments about the class were positive. However, there were some comments about the course feeling repetitive (we keep doing the same kind of activity over and over again) and people wondering what the point of this mathematics is. Both of those I will try to address in this Friday’s class. Not sure yet how as it’s only Tuesday. 🙂 Stay tuned.

Another thing that I have heard from students in this facility is that there are other mathematics courses that are being offered by a local community college, but they are often very introductory courses. Some students are ready for and are yearning for more advanced courses. I have inadvertently compounded this problem by adding yet another introductory course (in number theory), because I assumed that there were students who would not be ready to dive more deeply into mathematics. There definitely are students whose mathematical preparation is not sufficient for them to dive as deeply into this course as I originally intended, but then they are others who are ready for this course and more.

I think I will need to be rethink the course a bit. I will probably shift the course material away from the PCMI materials and incorporate other materials related to number theory. I will need to include more practical applications uses of the mathematics that we’re learning. And finally, I think I will need to more explicitly address the idea of mathematical competencies, perhaps by providing a partial list of the ways that students have been mathematically brilliant in the class.

An Inside-Out Course on Number Theory (Pt 2)

Link to previous post: Pt1

Teaching a math course inside of a prison is surprisingly unremarkable. Once we get into the classroom and starting doing mathematics, it’s easy to forget that you’re in a prison. We’re just a bunch of people learning mathematics together.

There are only several real complications I’ve encountered so far: (1) logistics, (2) limits on what we can learn about each other, and (3) much greater heterogeneity in my students’ prior experiences with and attitudes toward mathematics.

There are some significant logistical issues to figure out. My “outside students” and I have to drive from our campus to the prison each week (costs covered by Harvey Mudd College), make sure we’re dressed accordingly, don’t have cell phones or any other contraband, have proper identification, and follow any rules dictated by the prison.

There are some limits on what we can learn about each other. My outside students and I never ask for sensitive information from our inside students: the length of their sentences, what they did to wind up in prison, etc. It’s wonderful if an inside student choose to share that kind of information, but we don’t inquire. Both the inside and outside students are asked not to use their last names in this class. This is so that outside students aren’t tempted to look up information about the inside students, and the inside students don’t continue having a personal relationship with the outside students after the course is over (a rule of the Inside-Out Prison Exchange Program).

This past week, I was reminded about the great heterogeneity of mathematical experience that exists in our classroom. As you can imagine in any group of adults, there were are some people with positive attitudes toward mathematics, and some with negative views. Some of these stories came out during our first class meeting when we shared our experiences with each other about mathematics.

This past Friday, I got a bit of a shock. A student asked me, “What’s that little number next to that number?” He had not seen an exponent before. I was taken aback, not by his lack of experience with exponents, but that I had been so oblivious to that up to this point. There have been some exponents that have appeared in the course materials so far, but this question didn’t come up yet. We quickly talked about exponents and the student was fine and continued working. But, I was a bit shocked and still am a bit unsettled.

In the description for this number theory course, I wrote that some fluency with high school algebra would be required. It could be that some people are taking the class anyway, even if they aren’t fluent with high school algebra. Or it could be that some students have taken high school algebra in the past but have forgotten it. Either way, the reality is that there are students in this course that have very different experiences with mathematics. Some are just (re)learning about exponents and others are making sophisticated connections about what they’re learning. It’s my job as the instructor to make sure that we are all learning mathematics together regardless of our prior experiences; that is a pretty big challenge, however.

Other complications I’ve experienced so far are not germane to teaching Inside-Out. For example, if a student misses a class, I have to find ways to help them find out what they missed so that they can participate in class. My outside students and inside students have both missed classes for various reasons. Outside students have missed class due to illness or travel. Inside students sometimes cannot come to class because their dormitories are on “lock down” during our class.

There is one more significant difference between teaching courses at the Harvey Mudd and teaching through the Inside Out program. The pedagogy that we use in this course is quite different. As I mentioned in the first part of this post, I am not lecturing in this course. We do mathematics together through carefully sequenced problem sets that take into account what we learn during each class meeting and what students are wondering about. I do enjoy teaching in this way and wish that more of my Mudd courses could be taught this way. The pressure of having to “cover” a pre-determined body of material in a prescribed amount of time prevents me from fully teaching in this problem-based approach. One day, I would like to figure out how to teach in this humane way in my Mudd courses.

An Inside-Out Course on Number Theory (Pt 1)

This semester, I’m teaching a course entitled “Introduction to Number Theory” through the Inside-Out Prison Exchange Program. In short, what that means is that every Friday, I take a group of students from the Claremont Colleges (the “outside students”) with me to the California Rehabilitation Center to join 15 incarcerated students (the “inside” students) in learning some introductory topics in number theory.

The California Rehabilitation Center (CRC) is a medium-security state prison for men, located in Norco, California. Here’s a painting of CRC by Sandow Birk (held by the Pomona College Museum of Art).

“For [this] project I visited every one of California’s 33 state prisons and painted a picture of them. The idea was about the changes that California has gone through over its 150 years–from being seen in the 1850’s as an American Eden, where you could go west and dig out gold out of the ground and eat the oranges from the trees and it was always sunny and warm and you could strike it rich, to becoming the most incarcerated population on Earth. It’s shocking. And the more you learn about prisons the more nasty it all becomes.” — Sandow Birk

For over 20 years, the Inside-Out Prison Exchange Program, based at Temple University, has brought campus-based college students with incarcerated students for semester-long courses held in a prison, jail or other correctional setting all around the world. What I appreciate most about the organization is the way it approaches education as a collaborative endeavor and not one in which higher education professors and students go to a carceral organization to “help inmates” out of a sense of volunteerism or charity. Our local Inside Out program was started by Pitzer College and is run in part by a group of incarcerated men at CRC who make up our “Think Tank”. The truth is that I and the Claremont Colleges outside students are learning just as much as inside students are, if not more.

How are students selected? All students (inside and outside) are asked to fill out a questionnaire to find out why students want to take this course and what they hope to gain from the experience. There are several Inside Out courses that the Claremont Colleges offer each semester, and all of us instructors figure out how to allow the greatest number of students as possible to take our courses.

What are the goals of this course? While I do want students in this course to learn some interesting mathematics, the underlying goal of this course is for students to learn something about themselves and others through doing mathematics with each other. In particular, I am hoping that students in this class will have a more nuanced and complete understanding of what it means to be mathematically brilliant so that they can recognize that in themselves and others. This is one of the ways that I am hoping to create a rehumanizing mathematical experience for me and my students.

What is the course like? The Inside-Out Program is very particular about the kind of pedagogy we are to use. Lecture-based courses don’t provide for the kind of mutual engagement and co-learning that the program is trying to encourage. Therefore, I’ve structured my course using materials based off of my work with Bowen Kerins, Al Cuoco, Glenn Stevens in 2009 at the IAS/Park City Mathematics Institute Teacher Leadership Program.

On the first day, I tell students that this course is likely to be very different from any other mathematics course they’ve taken. The class is designed so that students learn from and with each other, not directly from me; I spend almost no time lecturing. Instead, the students work in small groups on a set of mathematical tasks during each class period. I’ve designed the tasks to pique curiosity and encourage students to make conjectures and look for patterns—in other words, the tasks are designed to engage students in doing mathematics the way that professional mathematicians do mathematics.

We basically spend almost all two hours of our time together doing math. I interrupt the work from time to time to facilitate students sharing their observations with each other. We close out the time by having a whole-class discussion and share-out about the (1) questions that we’re still wondering about, (2) interesting mathematical observations that we made, and (3) our gratitude toward one another for the contributions that they made to our learning.

Why number theory? Number theory is a wonderful area of mathematics that has a low threshold for entry and high ceiling for exploration. I have designed the course materials so that only experience with high school Algebra is required. Also, I am not at all an expert in number theory, so that allows me to approach things with a fresh perspective and to be surprised along with my students.

What’s it been like so far? We’ve already had four class sessions. We started by looking at the divisors of numbers and we’re currently thinking about modular arithmetic. Both the inside and outside students have been fantastic. Everyone seems to be deeply engaged in the mathematics and in working with each other.

Ideally, I would have had an equal number of inside and outside students, but right now I have 4 outside students and 15 inside students. We have been arranging ourselves in four groups of 4-5 students. This has worked out really well so far.

Unpredictable things happen all the time that prevent people from attending class. For example, during the first class session, parts of the prison were on lock-down so some students were not able to get to class. I have to be flexible and find ways to fold in students when they are able to attend class.

I hope to write more about my experiences throughout this semester. These are just some preliminary thoughts that I wanted to jot down.

This teaching and learning experience would not be possible without (1) the training and support I received in May 2018 from the Inside-Out Program, (2) support from administrators at the CRC, (3) the amazing students that are currently in the course, (4) and logistical support from the Claremont Colleges, made possible in part by a grant from the Andrew Mellon Foundation.

Next… Part 2

Research on class size and participation of women in undergraduate STEM courses

This 2019 paper by Ballen et al in BioScience entitled “Smaller Classes Promote Equitable Student Participation in STEM” is worth reading if you’re interested in equity, teaching, and STEM.

This large study (with 26 co-authors!) attempts to determine which features of undergraduate STEM classrooms correlate with more equitable vocal participation of women through a careful analysis of 5300 student-instructor interactions in 44 courses (observed over a full term) at six different institutions (4 in the U.S., 1 in Egypt, 1 in Norway). Answer: (1) class size was most impactful, followed by (2) the number of different strategies that the instructor used to elicit students’ vocal participation in class.

Educational literature up to this point has been very mixed about the connection between class size and student learning (see OECD 2012 report, CampbellCollaboration 2018 report, and many others). This is the first paper in which I’ve seen a strong argument for reducing class size: lowering class size increases the likelihood that women will participate in undergraduate STEM classes. In their data, increasing class size from 50 to 150 students decreased the likelihood of a woman participating by 50%.

The researchers were also able to reject several alternative hypotheses with their data. These alternative hypotheses included connections between equitable vocal participation and (1) abundance of student-instructor interactions per class period, (2) instructor gender, (3) proportion of women in the class, and (4) whether the STEM class was lower- or upper-division. (In other words, none of these things significantly correlated with more equitable participation.)

But since class size is often out of the instructor’s control, what can one do to make participation more equitable? The researchers found that instructors using a large repertoire of methods of eliciting vocal participation from students also got more equitable participation. (This makes me wonder, however, whether the size of an instructor’s repertoire of methods for eliciting student participation might correlate with instructor’s overall skill.) The article gives at least 7 different strategies that instructors can use to elicit students’ contributions in a classroom:

1) increase wait time between posing a question and selecting someone to answer in front of the whole class
2) using think-pair-share before selecting someone to answer
3) letting students work in small groups
4) having students write first before sharing out loud
5) soliciting multiple volunteers and calling on students only after a certain number of students have raised their hands
6) assigning student groups a number and using random number to select a group to answer
7) assigning a student in a group to be the “reporter” based on some arbitrary characteristic (e.g. random number or who woke up earliest)

I have a few more:
8) in addition to having good wait time, when posing a question to the class that you want students to respond to, select concise and clear wording for your question prompt, display the question on the screen (if possible) while reading it, and avoid that awkward continuous rephrasing of the question as a way of filling the silence
9) using vertical non-permanent surfaces (like white or chalk boards) to make student work visible before having students share ideas verbally to the whole class
10) positioning incomplete, partial, incorrect answers as being valuable to classroom discourse so as to lower the social risk for students to participate in whole-class discourse
11) using Google docs or equivalent online platforms to allow students to simultaneously contribute their ideas in an online space (this idea doesn’t involve vocal participation like the others, but it still involves students articulating their ideas in front of others)

Because the researchers’ posited reasons for women participating in class less than men involve imposter syndrome and social identity threat, that means these results should also to students of color in undergraduate STEM courses as well. In addition, I can see that many of the same arguments will also apply to secondary schools.

Characteristics of Effective Teacher/Faculty Professional Developent

Note: This blog post is based on a presentation that I gave at the 2019 MathFest in Cincinnati during a contributed paper session entitled “Professional Development in Mathematics: Looking Back, Looking Forward, on the Occasion of the 25th Anniversary of MAA Project NExT” organized by Dave Kung, Julie Barnes, Alissa Crans, and Matt DeLong.

For over 15 years, I have designed and led professional development for K-12 (mainly PCMI and MfA LA) and higher-education faculty (mainly Claremont Colleges Center for Teaching and Learning), primarily to help others enhance their teaching and learning.  I never received any formal training in how to do this kind of work, but I was fortunate to have worked alongside other educators in the field (Ginger Warfield, Gail Burrill, Peg Cagle, Pam Mason, and many others) and I have tried to learn as much as I can from the education literature.

Professional development for both K-12 and higher-education faculty is crucial if we want to continually improve the quality of education for our students and to reduce the loss of talent and resources that comes from faculty turn-over. And yet, anyone who has gone through professional development trainings and workshops knows that the main problem with professional development is that not all of it is good. In fact, some professional development is just plain awful. Bad professional development not only turns people off from wanting to continue to advance their skills, but it also muddies the waters about whether money spent on professional development is worthwhile.  However, the reality is that we are continually learning more about what effective teaching looks like and that information needs to be disseminated to teachers and faculty, so we will never stop needing good professional development for teachers and faculty. Moreover, “one constant finding in the research literature is that notable improvements in education almost never take place in the absence of [teacher] professional development” (Guskey, 2000, p. 4).

There is much less published research on the professional development of higher-education faculty than there is for K-12 teachers. This makes a lot of sense because there are many more K-12 educators than there are higher-education faculty and much more is spent on K-12 teacher professional development than for higher education faculty. I believe there is a lot that folks doing professional development for higher-education faculty can learn from what has been written about in the K-12 world.

Recently, I did an extensive literature search to find research on what effective K-12 teacher professional development looks like (not limited to mathematics). I found over 30 years of research commentaries, empirical studies, and meta-analyses that try to characterize effective professional development (Banilower, Boyd, Pasley, & Weiss, 2006; Birman, Desimone, Porter, & Garet, 2000; Blank & de las Alas, 2009; Borko, Jacobs, & Koellner, 2010; Darling-Hammond, Hyler, & Gardner, 2017; Desimone, Porter, Garet, Yoon, & Birman, 2002; Garet, Birman, Porter, Desimone, & Herman, 1999; Garet, Porter, Desimone, Birman, & Yoon, 2001; Little, 1993; Loucks-Horsley et al., 1987; Loucks-Horsley, Stiles, Mundry, Love, & Hewson, 2010; Stein, Smith, & Silver, 1999; Timperley, 2008; Timperley & Alton-Lee, 2008; Wilson, 2013).

There is a remarkable amount of consistency among all of this scholarship. To demonstrate this, I’ve selected three papers from the 16 papers listed above and summarized their lists of characteristics of effective PD below.

Loucks-Horsley, S., Harding, C. K., Arbuckle, M. A., Murray, L. B., Dubea, C., & Williams, M. A. (1987). Continuing to Learn: A Guidebook for Teacher Development. The Regional Laboratory for Educational Improvement of the Northeast and Islands; National Staff Development Council.

Characteristics of effective K-12 teacher professional development:

  1. Collegiality and collaboration
  2. Experimentation and risk taking
  3. Incorporation of available knowledge bases (by this they mean that teaching practice should be informed by research and validated in model programs and practices)
  4. Appropriate participant involvement in goal setting, implementation, evaluation, and decision making
  5. Time to work on staff development and assimilate new learnings
  6. Effective leadership and sustained administrative support
  7. Appropriate incentives and rewards
  8. Designs built on principles of adult learning and the change process (andragogy—the practice of teaching adult learners; includes opportunity to try new practices, guided reflection and discussion, time for significant change, balancing support and challenge)
  9. Integration of individual goals with school and district goals
  10. Formal placement of the program within the philosophy and organizational structure of the school and district (by this, they mean that it cannot be the effort of a few energetic individuals, it must be embedded in the organizational structure and culture)

Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What Makes Professional Development Effective? Results From a National Sample of Teachers. American Educational Research Journal, 38(4), 915–945.

Characteristics of effective K-12 teacher professional development:

  1. Focuses on subject-matter content and how students learn it
  2. Includes opportunities for teachers to become actively engaged in meaningful discussion, planning, practice
  3. Professional activities are coherently organized around goals that align with state and district standards and procedures
  4. More contact hours over a longer time span allows for learning to sink in
  5. Collective participation of people from the same school, department, or grade level is more helpful than participation of individuals from many different schools

Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective Teacher Professional Development (p. 76). Retrieved from

Characteristics of effective K-12 teacher professional development:

  1. Is content focused
  2. Incorporate active learning
  3. Supports collaboration
  4. Uses models of effective practice
  5. Provides coaching and expert support
  6. Offers feedback and reflection
  7. Is of sustained duration

I hope that you see many connections between the items on these three lists. And the same is true if you look across all 16 papers.

Based on this survey of the literature and my own experiences doing this work, here are a few important takeaway messages for people who lead and design professional development for both K-12 teachers and higher-education faculty. These ideas are oversimplifications, so you’ll need to think about how they might apply in your own context.

First of all, learning takes time and being able to see evidence of change takes even more time. We should not expect much to happen from a one-time 90-minute workshop. Programs that happen over longer periods of time are more likely to lead to real change in behaviors. This seems pretty obvious and yet a lot of professional development programs rely on the one-time workshop model.

Second, we need to make sure that the work that we are doing is aligned with the realities of the institutional (schools, districts, colleges, universities) and departmental contexts faced by participants in our programs.

Third, authentic community is important because it supports collaboration. Having some shared context is one way to create an authentic community.

Fourth, program evaluation is crucial. This is not an item that we see in the lists above, but it is one that I have found to be true based on the work I’ve done so far. Effective professional development efforts are ones that can document growth and success over time, both for ourselves and for our stakeholders and potential funders. That documentation requires us to be strategic about program evaluation and assessment. We have to get much better and smarter at how we evaluate our programs. Not everything can be quantified, but we can’t let the challenge of measuring progress keep us from constantly improving through program evaluation..

I don’t think these characteristics are necessary and sufficient conditions for professional development to be effective. I suspect they are only necessary at best. There are probably other conditions that are required too.

What do you think are essential characteristics for teacher/faculty professional development to be effective?


  • Banilower, E. R., Boyd, S. E., Pasley, J. D., & Weiss, I. R. (2006). Lessons from a Decade of Mathematics and Science Reform: A Capstone Report for the Local Systemic Change through Teacher Enhancement Initiative. Retrieved from Horizon Research, Inc. website:
  • Birman, B. F., Desimone, L., Porter, A. C., & Garet, M. S. (2000). Designing Professional Development That Works. Educational Leadership, 57(8), 28–33.
  • Blank, R. K., & de las Alas, N. (2009). Effects of teacher professional development on gains in student achievement: How meta-analysis provides evidence useful to education leaders. Washington, DC: Council of Chief State School Officers.
  • Borko, H., Jacobs, J., & Koellner, K. (2010). Contemporary approaches to teacher professional development. International Encyclopedia of Education, 7(2), 548–556.
  • Darling-Hammond, L., Hyler, M. E., & Gardner, M. (2017). Effective Teacher Professional Development (p. 76). Retrieved from Learning Policy Institute website:
  • Desimone, L. M., Porter, A. C., Garet, M. S., Yoon, K. S., & Birman, B. F. (2002). Effects of Professional Development on Teachers’ Instruction: Results from a Three-Year Longitudinal Study. Educational Evaluation and Policy Analysis, 24(2), 81–112.
  • Garet, M. S., Birman, B. F., Porter, A. C., Desimone, L., & Herman, R. (1999). Designing Effective Professional Development: Lessons from the Eisenhower Program [and] Technical Appendices (No. ED/OUS99-3). American Institutes for Research.
  • Garet, M. S., Porter, A. C., Desimone, L., Birman, B. F., & Yoon, K. S. (2001). What Makes Professional Development Effective? Results From a National Sample of Teachers. American Educational Research Journal, 38(4), 915–945.
  • Guskey, T. R. (2000). Evaluating Professional Development. Corwin Press.
  • Kennedy, M. M. (2016). How Does Professional Development Improve Teaching? Review of Educational Research, 86(4), 945–980.
  • Little, J. W. (1993). Teachers’ professional development in a climate of educational reform. Educational Evaluation and Policy Analysis, 15(2), 129–151.
  • Loucks-Horsley, S., Harding, C. K., Arbuckle, M. A., Murray, L. B., Dubea, C., & Williams, M. A. (1987). Continuing to Learn: A Guidebook for Teacher Development. The Regional Laboratory for Educational Improvement of the Northeast and Islands; National Staff Development Council.
  • Loucks-Horsley, S., Stiles, K. E., Mundry, S., Love, N., & Hewson, P. W. (2010). Designing Professional Development for Teachers of Science and Mathematics (Third).
  • Stein, M. K., Smith, M. S., & Silver, E. (1999). The development of professional developers: Learning to assist teachers in new settings in new ways. Harvard Educational Review, 69(3), 237–270.
  • Timperley, H. (2008). Teacher professional learning and development (No. Educational Practices-18; p. 32). International Academy of Education.
  • Timperley, H., & Alton-Lee, A. (2008). Reframing Teacher Professional Learning: An Alternative Policy Approach to Strengthening Valued Outcomes for Diverse Learners. Review of Research in Education, 32(1), 328–369.
  • Wilson, S. M. (2013). Professional Development for Science Teachers. Science, 340(6130), 310–313.

Effect of Vertical Non-Permanent Surfaces (VNPS) on Task Cognitive Demand: A Case for Pedagogical Reasoning

This blog post has to do with vertical non-permanent surfaces (VNPS), but it really is the need for more pedagogical reasoning and sense-making in our community. I don’t mean to pick on VNPS in particular–I am using it as an example only because I happen to have some data about it.

About VNPS: (I apologize for using the acronym VNPS, but “vertical non-permanent surfaces” is lot of typing.) In a nutshell, this instructional strategy involves having students working at a whiteboard/blackboard (mounted on wall or easel) or nifty erasable plastic sheets that stick to the wall instead of working at their desks. The instantiation of this instructional strategy can vary: students can be asked to work solo or in small groups; all the students might be working on the same task or different tasks; the teacher might have students stay at their stations the whole time or combine the activity with a gallery walk. There are endless variations. Search Twitter to learn more. VNPS can bring lots of wonderful benefits to a classroom, and I’m not disputing those things here. You can also find some research (here’s one example) on VNPS.

In this blog post, I would like to make the argument that VNPS makes student thinking visible in the classroom and sometimes that visibility can lower the cognitive demand of the task. This likely happened in some classrooms that I visited a few months ago.

Information about the classrooms:

  • Four different teachers at the same school were implementing the same mathematical task in an Algebra 1 class and all asked students to work at whiteboards in groups of 2 or 3. The four teachers work closely together and decided on the task and implementation of the task via VNPS together.
  • This task was given near the beginning of the school year, when students were beginning to develop their understanding of linear growth.
  • I observed four periods on a day when the periods were about 50-minutes long. The teachers had students working at the board roughly between 15 to 30 minutes.
  • I should also admit my own bias here in that I know these four teachers well and respect them highly. I look to them to gain insights about teaching mathematics.

Here is the task that students were working on.

Person A starts off with 400 widgets and gains 20 new widgets every day. Person B starts off with 50 widgets and gains 70 new widgets every day.

1) For each of the next 10 days, who will have more widgets?

2) Will there be a time when both people have the same number of widgets?

3) Is it possible for them to have the same number of widgets again past the ten day period you have explored? Explain.

Make sure that all of your answers are thoroughly explained and justified on your boards. Use any and all mathematical methods/models you can think of to explain your answers and reasoning.

Overall, the quality of the student work was very high. All four teachers were careful to avoid telling students to use specific procedures (to use a graph, or a table, etc.) or to “help” students in a way that lowered the cognitive demand of the task.

(Modified version of task to protect anonymity of the teachers and students involved. The original task had a context that was much more interesting that “widgets.”)

As you might expect for this open-middle task, students used a variety of approaches for the task including tables, graphs, and symbols. Across all four of these Algebra 1 classes, I observed 39 different student groups. Out of the 39 groups, 31 of them displayed a table of numbers similar to these:

Some tables listed number of widgets for A and B separately…
… while some students created one table for both A and B together. Notice this group above circled the day in which both A and B had the same number of widgets and used that to answer Question #1.

Out of the 39 groups, 11 of them displayed a graph similar to these:

Some students created graphs for A and B separately…
.. while two groups graphed A and B together, showing the point of intersection more clearly.

Out of the 39 groups, 16 of them showed some form of algebraic expression. Eight groups set up an equation and solved for the day when the number of followers was the same. Here’s a table summarizing the frequency of various approaches that students used.

What strategies were used in all four class periods? This table shows the number and percentage of groups in each class that attempted a particular strategy, based on observations of students’ work during class.

Tables were the most common approach taken up by students. That is not surprising given that it was the beginning of the year and not students might have been fluent with graphing quantities and using algebraic expressions yet. In addition, the wording of question 1 led to most students to start off by listing the number of widgets by days.

Algebraic approaches were the least common. However, Period 3 stands out in that 5 out of 8 groups used an algebraic approach that also involved solving an equation. Here is the student work for those 5 groups. (As with the other photos here, I’ve edited the photos slightly to protect the identities of the students and teachers.)

Algebraic approach #1
Algebraic approach #2
Algebraic approach #3
Algebraic approach #4
Algebraic approach #5

Did you notice how all five examples of student work from this class period bear a striking resemblance to each other?

  • All of the student groups used “x” as the independent variable. There were other examples from other classes where the variable used was “d” (for days). Similarly, when a dependent variable was used, it was always “y” even though the objects being counted in this problem didn’t suggest the letter “y”.
  • All of the student groups solved for the variable using the strategy of adding or subtracting quantities from both sides of the equations at first. And, they all showed their work in the same way. Four of the groups chose to subtract 20x as the first step.

Based on these observations and my own notes taken on what students were doing during this period, I am almost certain that students looked around the room and copied this algebraic approach to their board. One group first came up with the approach, then I observed other groups copy it from the first group and from each other. It was fascinating to watch how ideas spread throughout the room in this and other periods.

(By the way, the teacher of this Period 3 class also communicated that a significant proportion of students in that class all had the same very demanding 8th grade teacher the previous year. By all accounts, that teacher apparently had remarkably inflexible standards for how to “do math” and reinforced them using methods such as public humiliation. We wonder how much of the very similar boardwork from this period came from having had this particular approach to solving equations drilled into students the previous year, and how much a fear of public humiliation might have contributed to students copying each other.)

Evidence of students getting ideas from each other in other periods was not as strong, but I believe it also happened to some extent. For example, I observed students who saw that another group had a graph and so that led them to also include a graph. It is also interesting to me that all of the groups never had any disagreement on whether the number of widgets was 400 and 50 on day 0 or day 1. And, the fact that there were no arithmetic errors at all across all 39 work samples leads me to suspect that perhaps were checking their numbers with each other.

It seems like a pretty obvious point to make that if you’re going to have students working publicly on a task using VNPS, you might get students getting ideas or directly copying work from each other. But, I think this is an important point to make for several reasons.

First of all, the possibility of students’ revealing ideas to their peers during the enactment of a mathematical task, voluntarily or involuntarily, means that the use of VNPS has the potential to affect the cognitive demand of that task. The QUASAR project has helped our field understand how the cognitive demand of mathematical tasks tends to decrease as they are successively taken up by teachers then students.

A representation of how a math lesson unfolds during classroom instruction. Taken from “Implementing Standards-Based Mathematics Instruction: A Casebook for Professional Development” (1st ed, 2000) by Mary Kay Stein, Margaret Schwan Smith, Marjorie A. Henningsen, Edward A. Silver.

In this case, all four teachers were very careful not to lower the cognitive demand of the task through their interactions with students during instruction. However, they might not have anticipated that the cognitive demand might have been affected by the choice of using VNPS.

Second, the possibility of students being able to see each other’s work during a mathematical task has implications for whose work gets valorized and ignored in the classroom. Imagine a hypothetical situation in which Students A, B, C are working together in a small group. And, imagine that Student X (not in their group) is a student that most others in the class perceive (accurately or not) to be mathematically competent. Student A shares a good idea with Students B and C, but they ignore her contribution and defer to Student X’s work instead, which they spy on from across the room. This hypothetical example is one possible way in which the public sharing of student work in progress might affect whose ideas get taken up.

Borrowing the language of complex instruction: VNPS could exacerbate any status issues that you might have in your classroom because interactions with any high status students are no longer restricted to the small group that those high status students belong to.

Third, the possibility of students being able to gain ideas from each other by looking around the room limits a teacher’s ability to use tasks to assess students’ abilities and understandings. Let’s return to the implementation of the widgets task described above. If the Period 3 teacher above wanted to know which solution strategies for modeling linear growth students remembered from their 8th grade math class, the VNPS enactment of this task might have led to an erroneous conclusion that many of them could proficiently work with algebraic expressions and solve equations.

I am not criticizing these four teachers for lowering the cognitive demand of the task. And in fact, there is one aspect of their use of VNPS that I think is quite brilliant: Because they did not specify whether to use tables, or graphs, or algebraic expressions, the fact that students had different approaches and then copied them to their work areas means that they had opportunities to make connections between these different representations of the scenario. And, I observed these teachers encouraging students to make these connections when they talked to individual student groups.

The purpose of highlighting these issues with VNPS is also not to demonize this very useful instructional strategy or to criticize these four teachers. Rather, I highlight these subtleties with VNPS to illustrate a more general point: when we educators talk about our teaching with each other, we tend to focus on the “hows” and “whats” rather than the “whys” and “whens”, and that incomplete transmission of ideas limits the effectiveness of those conversations.

A lot of what gets presented at conferences, written about in the Math Twitter Blog-o-Sphere, including a lot of what I have written myself, focuses on the “hows” and “whats”: how I create performance tasks when I use standards-based grading, what formulas I use in my grading scheme, what I do to handle re-dos, etc. What gets talked about less is why I use standards-based grading in the first place, when might I use that strategy over a different strategy, etc. We tend to focus on the “hows” and “whats” because that practical information is, um.., practical, so it helps us implement ideas quickly. However, the “hows” and “whats” likely don’t help us understand why something doesn’t work the way we thought it would, or how we adapt an idea to fit within our specific teaching contexts.

To effectively spread instructional innovations across a community, members of that community need to not just explain how to do X, but why they do X, when they do X instead of Y, what aspects of their teaching contexts lead them to implement X in the specific way that they have.

For example, when we talk about VNPS, let’s also talk about why we use VNPS as opposed to other instructional strategies. We shouldn’t use VNPS just because they are trendy–there should be some specific instructional purpose behind those decisions.

VNPS can be an effective strategy for getting students out of their seats and actively engaged in mathematical tasks with each other, but you could do something similar by setting up stations around the room and having manipulatives or handouts on desks as opposed to having mathematical work shared publicly around the room. What are the affordances of each in your specific context?

VNPS can be an effective way to get students to try out incomplete ideas in draft form because you can erase your work easily. However, you can achieve something similar by having personal whiteboards or shared whiteboards placed horizontally on tables. If you want student work to be shared publicly, that can still be done after the work is completed by having the personal whiteboards lifted up or placed along the wall for a gallery walk.

The specific arrangement of your classroom also matters quite a lot. One of the four teachers I mentioned above has a rather large room and even though students were working on white boards mounted around the room, they were far apart from each other that the only transmission of ideas that I observed were by nearest neighbors. In other teachers’ rooms, white boards were packed closely together and it was much easier for students to see each others’ work.

So, instructional goals, classroom culture, students’ predilections, even furniture arrangement all matter a great deal when we talk about our teaching ideas with each other. We should aim to situate our conversations about teaching innovations more thoroughly so as to enable others to make the greatest use of those ideas. If we don’t, then we increase the risk of unexpected consequences when we try out each others’ ideas and we miss out on opportunities to strengthen our own and others’ pedagogical reasoning and sense-making skills.

Think about it this way: if a student tells you the answer is 42, you will probably also ask them to justify their answer. Why then, should we as educators, only tell each other the things we’re doing in our classrooms but not also justify our ideas with pedagogical sense-making and reasoning?

What Must Teachers Possess to Support Their Equitable and Inclusive Teaching?

What must learners of mathematics possess so that they can learn mathematics on their own? For example, once a student graduates from my college, what does she need to continue learning mathematics on her own? Besides having some body mathematical content knowledge, this student would probably need to have some mathematical habits of mind, some productive dispositions about learning mathematics, a sense of self-efficacy, a growth mindset about learning mathematics, perseverance, etc.

So, now let’s ask an analogous question: say we are trying to help teachers (of all levels, primary, secondary, tertiary) learn how to teach in more equitable and inclusive ways. What must they possess to do that? They might need to know some instructional strategies for equitable and inclusive teaching, but clearly they need more than that.

Why is it, then, that most professional development is focused on helping teachers acquire instructional strategies, but not much more than that? It would be as silly as if we only focused on helping students acquire content knowledge in our fields and none of the other things that they might need to continue their own learning.

To all of you who think deeply about equity and justice in education and are looking for ways to help teachers teach in more equitable and inclusive ways, here’s an important question: What must teachers possess to continue making their classrooms more equitable and inclusive long after they have left our workshops or training sessions?

Clearly they need some of the “whats” and “hows” of equitable and inclusive teaching–they need to learn some equitable instructional strategies. Pick your favorite ones: using “equity sticks” to call on students more equitably, finding ways to remove gendered language and examples from your course content, active learning strategies (like inquiry-based learning, POGIL, etc.), justice-oriented curricular materials, etc. Those things are important, but what else?

With input from Sumun Pendakur and Peg Cagle, I’ve compiled this list of things that teachers need to continue making their classrooms more equitable and inclusive. Will you help me edit/add to this list?

  1. A Sense of Purpose: What’s the force that compels me to do more and grow? It could come from a personal moral and ethical stance, spiritual practice, determination to make the world better for future generations, or affinity with your institution’s mission, etc.
  2. Productive Dispositions about Teaching and Equity:
    • (a) A belief that time and effort I spend on improving teaching and learning can result in fruitful changes in my classes that benefit students
    • (b) Similar belief that time and effort spent learning about equity and inclusivity can benefit my classes and students
  3. Humility: Knowing that there is something that I can learn from everyone and that I will never really “arrive” at the perfection in my teaching but that I still strive toward that ideal
  4. Growth Mindset about Equity: Believing that my capacity to be more equitable and inclusive in everything that I do can improve with effort, rather than insisting that people are just “good” or “bad” or “racist” or “woke” (I wrote about this in a previous post.)
  5. Perseverance: Having the courage to continue trying to teach in better, or more equitable and inclusive ways even if I don’t attain success the first few times
  6. Equity-Oriented Habits of Mind:
    • (a) Having an asset-oriented, as opposed to deficit-oriented, way of thinking about students. Knowing that I and my institution have the responsibility to help students succeed and that there are things within my control that can make a difference
    • (b) Asking not just about the “whats” and “hows” of teaching, but also the “whens” and the “whys”; having an understanding that classrooms and people are too complex to boil things things down to “best practices” and instead being able to think of the classroom as an ecology of people, environments, and their relationships with each other.
    • (c) Having a habit of self-reflection about one’s teaching practice.
  7. Community: Participation in a community with other professionals who are struggling through similar problems of practice helps me continue to grow. But it’s not just having folks around, but also knowing how to engage in effective professional conversations. Julianne Vega recommends the book Talk About Teaching! Leading Professional Conversations by Charlotte Danielson.
  8. Joy and Humor: This kind of work is too difficult to sustain without self-care. The ability to find joy in your work and to take work seriously, but not take yourself too seriously, helps us pick ourselves up when we make mistakes and continue learning. I also highly recommend the “Killjoy Survival Kit” in Living a Feminist Life by Sara Ahmed.

There is a great deal of overlap here between things one might need to improve as a teacher in general and things one might need to teach in more equitable and inclusive ways. Items #4 and #6a above seem to me to be specific to teaching in more equitable and inclusive ways.

What else is missing from this list? And the big question, of course, is: how do we cultivate these things in ourselves and help others cultivate them too?